Prime-product coefficient conjecture for Gröbner bases

Let $\mathbf{k}$ be a field of characteristic zero, let $R=\mathbf{k}[x_1,\ldots,x_n]$, and let

$$I_{a_1,\ldots,a_{n+1}}=(x_1^{a_1},\ldots,x_n^{a_n},(x_1+\cdots+x_n)^{a_{n+1}}).$$

Fix the reverse lexicographic term order and consider the Gröbner basis of this ideal.

Prime-product coefficient conjecture. Each coefficient occurring in the Gröbner basis of $I_{a_1,\ldots,a_{n+1}}$ is a possibly empty product of primes less than $\max\{a_1,\ldots,a_n\}$.

The claim is motivated by computer experiments. The source verifies the special case $a_1=\cdots=a_n=2$ through the paper's theorem, but gives no resolution of the general conjecture.