CHICAGO — When Elon Musk agreed this summer to build a futuristic tunnel to Chicago’s busiest airport on his own dime, many Chicagoans voiced doubt.

The idea sounded fanciful: Electric pods hurtling under the city at 150 miles per hour. And experts questioned whether Musk’s company could dig a 17-mile tunnel in a few years.

Then Mayor Rahm Emanuel, the tunnel’s main supporter, stunned Chicago this month by announcing that he would not run for a third term, casting more doubt over the project’s future. A few days later, Mr. Musk, the chief executive of the electric-car maker Tesla, came under scrutiny for apparently smoking marijuana during an interview. The episode followed a summer of turmoil that led to the resignation of the company’s chief accounting officer and raised questions over Mr. Musk’s emotional state.

Mr. Emanuel and Mr. Musk insist the project is moving forward and technically feasible, but some transportation experts say it is unlikely to happen, and if it does, will take longer than promised. Even if Mr. Musk’s engineers can dig a tunnel faster than other companies, his pod system, which the entire project is based on, is still a work in progress.