Is it absolutely necessary to keep caverject refrigerated? My version of it is tri-mix forte?

If it in the powder form it can be stored at room temperature for 3 months.

Otherwise these are the storage instructions for Caverject.

Store in the refrigerator. Do not freeze.

You may store the suppositories in the refrigerator, but do not freeze them.

Keep out of the reach of children.

Do not keep outdated medicine or medicine no longer needed.

Alprostadil for Injection while in the powder form can be stored at room temperature (between 15 and 25 °C or 59 and 77 °F) for 3 months. After it is mixed, the solution must be used immediately. Suppositories may be stored at room temperature.

https://www.drugs.com/cons/caverject.html