Incomplete Datta-Ghorpade conjecture for maximum projective point counts

Let $e_r(d,m)$ denote 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 $H_r(d,m)$ be the affine point-counting quantity defined from the lexicographically ordered exponent vectors in $\Omega(d,m)$. Let $\pi_k=|\mathbb{P}^k(\mathbb{F}_q)|$. Incomplete Datta-Ghorpade conjecture. Given $m,d\geq 1$, $1\leq r\leq\binom{m+d-1}{d-1}$, and $q\geq d+1$, one has

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

This conjecture was proposed after the Boguslavsky-Tsfasman conjecture was disproved in part of its stated range. The source reports proofs for $d=1$, for $m=1$, and for additional bounded ranges of $r$, while presenting the conjecture as superseded by a complete extension covering all $r$; its resolution status is therefore not established by the supplied text.