Initial-ideal decomposition conjecture for almost complete intersections

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

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

Let $J_{a_1,\ldots,a_{n+1}}$ be the ideal generated by all monomials $x_1^{b_1}\cdots x_n^{b_n}$ in $I_{a_1,\ldots,a_{n+1}}$ such that $b_1<\min\{a_1,a_{n+1}\}$ and $b_j<a_j$ for every $j>1$.

Initial-ideal decomposition conjecture. The initial ideal decomposes as

$$\operatorname{in}(I_{a_1,\ldots,a_{n+1}})=(x_1^{\min\{a_1,a_{n+1}\}},x_2^{a_2},\ldots,x_n^{a_n})+J_{a_1,\ldots,a_{n+1}}.$$

This is proposed as a generalization of a proposition proved in the paper; the source gives no resolution status.