The United States and the European Union have expressed serious concern at the latest deadly fighting around Nagorno-Karabakh, saying that the parties to the Armenian-Azerbaijani conflict should urgently take measures to bolster the ceasefire regime there.

“The United States is deeply concerned over the recent violations of the ceasefire, which have resulted in multiple casualties on the Line of Contact over the last week,” the U.S. State Department spokeswoman, Heather Nauert, said in a statement issued late on Wednesday.

“We urge the sides to avoid escalation and encourage immediate consideration of measures to reduce tensions along the Line of Contact and the international border between Armenia and Azerbaijan,” she said.

“The European Union expects de-escalation and restraint in deeds as well as in words,” read a separate statement released by an EU spokesperson on Thursday.

The EU statement came shortly before Karabakh’s Armenian-backed Defense Army claimed to have killed four Azerbaijani soldiers while thwarting an overnight commando attack on its frontline positions at an unspecified section of “the line of contact.” It said its troops suffered no casualties.

There was no immediate reaction to the claim from the Azerbaijani military.

The Karabakh Armenian army promised a “targeted and disproportionate” retaliation after three of its soldiers were killed by Azerbaijani forces on June 16. Another Armenian soldier was killed at a different frontline section the following morning.

Earlier on June 16, the Azerbaijani Defense Ministry reported that one of its soldiers was shot dead by the Armenian side.

The warring sides blamed each other for the escalation that coincided with the latest visit to the conflict zone by U.S., Russian and French diplomats co-chairing the OSCE Minsk Group.

In a joint statement issued at the end of their regional tour on Monday, the three mediators said they “appealed to the leadership of Azerbaijan to avoid further escalation.” “The Co-Chairs are sending the same message to the leadership of Armenia and de facto authorities of Nagorno-Karabakh,” they added.

The co-chairs also urged the sides to take confidence-building measures that would reduce tensions on the frontlines. The U.S. and EU statements called for “immediate consideration” of such safeguards.

The Armenian and Azerbaijani presidents agreed to let the OSCE deploy more field observers in the conflict zone and work out a mechanism for investigating truce violations there at their face-to-face talks held a year ago. Baku has been reluctant to implement those agreements, saying that they would cement the status quo in the absence of progress towards a resolution of the Karabakh conflict.

The U.S. and the EU also echoed the mediators’ calls for the conflicting parties to “re-engage in negotiations on substance, in good faith and with political will.”