Residue distribution conjecture for the fundamental domain of Pascal's triangle modulo p

Let $p$ be a prime, and let $A_p(r)$ denote the number of occurrences of the residue $r$ in the fundamental domain of Pascal's triangle modulo $p$.

Residue distribution conjecture. As the prime modulus $p$ tends to infinity,

$$A_p(1)\sim 3p,\qquad A_p(-1)\sim p,$$

and, if $r\neq -1,0,1$, then

$$A_p(r)\sim \frac{p}{2}.$$

The fundamental domain consists of the entries $\binom{n}{m}\bmod p$ with $n,m<p$. The conjecture predicts an approximately uniform distribution among nonzero residue classes, apart from the exceptional classes $1$ and $-1$; it is attributed in the source to the cited work of 2019Fract..2750098B.