The $k$-ary rank conjecture for monomials

Let $F=x_1^{d_1}\dotsm x_n^{d_n}$, where $0<d_1\leq\dotsb\leq d_n$, and let $r_k(F)$ denote the least number of summands in a decomposition of $F$ whose summands are forms depending essentially on at most $k$ variables. The bounds in the source give

$$r_k(F)\leq (d_2+1)\dotsm(d_{n-k+1}+1).$$

The $k$-ary rank conjecture. The upper bound is an equality:

$$r_k(x_1^{d_1}\dotsm x_n^{d_n})=(d_2+1)\dotsm(d_{n-k+1}+1).$$

This is known for $k=1$, for $k=n$, and whenever $d_1=\dotsb=d_{n-k+1}\leq d_{n-k+2}\leq\dotsb\leq d_n$; the general case remains open in the source.