The unbounded multiplicity conjecture for differences of tetrahedral numbers

Let $N$ be a positive integer. A positive integer solution of the equation below is a pair $(n,m)$ of positive integers.

Unbounded multiplicity conjecture. For each $N\in\mathbb{N}$ there is $d_N\in\mathbb{N}$ such that

$$\binom{n}{3}-\binom{m}{3}=d_N$$

has at least $N$ positive integer solutions.

This conjecture arises from computational examples showing that fixed differences can have several representations as differences of binomial coefficients. It predicts that the number of such representations is unbounded, but no proof or disproof is given here.