Complete Beelen-Datta-Ghorpade conjecture for projective point counts

Let $e_r(d,m)$ be the maximal number of $\mathbb{F}_q$-rational points of a projective algebraic set defined by $r$ linearly independent homogeneous degree-$d$ polynomials in $m+1$ variables. Let $H_j(d-1,m-l+1)$ be the affine point-counting quantity defined from the lexicographically ordered exponent vectors, and let $\pi_k=|\mathbb{P}^k(\mathbb{F}_q)|$. Complete Beelen-Datta-Ghorpade conjecture. Suppose that $m,d\geq 1$, $1\leq r\leq\binom{m+d}{d}$, and $q\geq d+1$. Let $l$ be the unique integer with $1\leq l\leq m+1$ such that

$$\binom{m+d}{d}-\binom{m+d+1-l}{d}<r\leq\binom{m+d}{d}-\binom{m+d-l}{d}.$$

Set

$$j=r-\binom{m+d}{d}+\binom{m+d+1-l}{d},$$

so that $0<j\leq\binom{m+d-l}{d-1}$. Then

$$e_r(d,m)=H_j(d-1,m-l+1)+\pi_{m-l}.$$

This extends the incomplete Datta-Ghorpade formula from a restricted range of $r$ to all admissible $r$. The source attributes the extension to Beelen, Datta, and Ghorpade and does not state a resolution, so the conjecture remains open here.