An inequality involving common polynomial divisors of maximal degree

Question

Let $R$ be a domain. For all $f,g\in R[X]$ non-zero, define $$D(f,g):=\max\{\deg d:d\in R[X],d\mid f,g\}.$$ Question: For which $R$ does the inequality $$D(f,g)+D(f,h)\ge D(f,gh)$$ hold for all $f,g,h\in R[X]$ non-zero? In particular:

1. Is it enough for $R$ to be pre-Schreier?
2. Is it enough for Euclid's lemma (the general version, $a|bc$, $(a,b)=1$, then $a\mid c$) to hold in $R$?

What I know so far, is that the inequality holds for all $f,g,h\in R[X]$ non-zero if $R$ is a GCD domain.