It has been two weeks since Vittorio Missoni, his companion, and two close friends took off from the tourist island of Los Roques headed toward Caracas’s Simon Bolivar International Airport on the Venezuelan mainland. The foursome had been on a fishing expedition over the holidays and were headed back to Italy to start the new year. Missoni was expected to attend men’s fashion week in Milan this week, but his plane disappeared from radar shortly after taking off from the island.

At first glance, the disappearance seems like a standard air-to-sea accident, but the Missoni family is not yet ready to give up hope. “A plane cannot vanish in this way, on a short route, without leaving any trace,” Missoni’s 28-year-old son Ottavio told an Italian newspaper last week. “I remain convinced that the least plausible reason is that they crashed into the water.”

For the family, there are just too many discrepancies to accept what seems to be the obvious fate of the foursome and their crew. Because of the lightweight nature of the aircraft, it would have broken up on impact, leaving a debris field of floating fiberglass and, potentially, personal effects—at the very least floating seat cushions or suitcases—none of which have been found, despite an extensive air and sea search.

But more disturbing is a series of cellphone anomalies. On Jan. 6, according to Italian wire service ANSA, more than 48 hours after the plane disappeared, the cellphone belonging to fellow passenger Guido Foresti sent a message to Foresti’s son indicating that the phone was back in range after being out of that zone since earlier that day. Calls made later to both Foresti and his wife’s number indicated that the phones were off. A day later, calls to Foresti’s wife’s phone rang 10 times before automatically transferring through to the phone’s answering service, indicating that her phone was also momentarily on or back in cell-tower range. According to several Italian newspapers, a list of calls registered by the local Venezuelan telephone carrier the Italians’ phones were roaming through showed that both the Foresti phones made a series of calls at noon on Jan. 4, several hours after the plane disappeared.

Three days after the plane disappeared, Missoni’s sister Angela told reporters in Milan that she thought the foursome had been kidnapped “It’s better to be kidnapped than at the bottom of the sea,” she said. “We hope all four are alive and well. We also hope that the searches go in all directions.” The Italian glossy Oggi followed up on the kidnapping theory, reporting on Thursday that in the last 15 years in Los Roques, more than 57 small airplanes have disappeared with out a trace, mostly tied to alleged drug-running. Citing Hedy Ramirez, secretary of security for Venezuela’s state of Guarico, nearly half of the planes were recovered in a sting operation in February 2010. The Missoni family is clinging to hope that their company chief’s plane might have been involved in a similar incident.

Still, there is just as much information that points to a fatal crash. Italian air-safety authorities joined the Venezuelan search five days after the disappearance and were startled to discover that Transaereo 5074, the company that owned and operated the Missoni plane, didn’t actually have the proper certification to fly the tourist route, even though it had been operating for years. A representative from the company told the Associated Press in Caracas that they had indeed met all the safety requirements, but were waiting on the long bureaucratic process of certification. The Italian authorities also originally reported that the 72-year-old pilot didn’t have a valid license. They said his annual medical fitness test, required for all pilots over the age of 65, had not been renewed for two years, but it was later discovered that the renewal was valid and his license was good through May 2013. There was a co-pilot on board the aircraft, but he was sitting in the forward passenger seat, not the pilot’s seat when the plane took off. Venezuelan authorities countered that the lack of license or certification was not necessarily relevant in the investigation. “These are not yet considered factors directly related to the dynamics of the event,” the Venezuelan air authority said in a statement. “They are only facts in an investigation."

According to a blip on a Venezuelan air-control radar, the last known location of Missoni’s plane was just 11 miles from the Los Roques airport. Based on that location, the trajectory of the small twin-engine Norman BN-2A would have crossed an area of the Caribbean that is too deep for standard search-and-rescue instruments, which may be why it hasn’t been found yet. The Italian embassy in Caracas issued a statement explaining that it was extending the standard eight-day window on search-and-rescue missions as it waited for deep-sea equipment to reach the area. “It is an area where the depths reach 2,000 meters, too deep for the instrumentation available at the moment,” according to a statement from the Italian embassy in Caracas obtained by Newsweek and The Daily Beast. To placate the Missoni family, they want to be sure the plane went down, even if recovery is impossible due to the depth of the waters.

Yet Venezuelan authorities say the plane is indeed at the bottom of the sea, due to either mechanical failure or bad weather. For obvious reasons, they don’t want to propagate the drug-related kidnapping subplot, which the Venezuelan coast guard has dismissed as a “conspiracy theory.” But for the families of Missoni and his traveling companions, until they find a debris field or even a tiny trace of the missing plane, they say they will hold on to hope that their loved ones are alive. “We want to hope,” Missoni’s sister Angela said. “Because for us it is important.”