Yes, a player can complete or trigger conditional or triggerable missions before discarding down to four cards during the end-of-round step, as long as they do so as a free action on the turn in which they pass.

Explanation

• According to page 19, "Passing is a main action – which means you can do free actions on the turn in which you pass. Then resolve these steps in order, without playing any more free actions: 1. Discard down to 4 cards if you have more than 4."
• This means that after you declare you are passing (your main action), you may still perform any free actions, including completing or triggering missions, before you proceed to the discard step.
• On page 15, it is stated that "At any time during your turn, if the condition is met, you can complete the mission as a free action..." and "It is possible to trigger some missions even outside of your turn."
• However, once you begin the end-of-round steps (starting with discarding down to four cards), you may not perform any more free actions.

Summary

A player may complete or trigger conditional or triggerable missions as free actions after passing but before discarding down to four cards. Once the discard step begins, no further free actions are allowed.

Source: page 19 (Passing and end-of-round steps), page 15 (Missions and free actions)

Source: Core Rulebook (pp. 15, 18, 19, 26, 7)