The arithmetic-progression conjecture for very triangular numbers

Let $t_j=j(j+1)/2$ be the $j$th triangular number, let $VT$ denote the set of very triangular numbers, and let $AP(k)$ denote an arithmetic progression of positive integers of length $k$.

Very triangular arithmetic-progression conjecture. For any positive integer $k$, there exists an arithmetic progression $AP(k)$ of length $k$ such that $t_j\in VT$ for all $j\in AP(k)$.

This is the very triangular analogue of the Green–Tao phenomenon for arbitrarily long arithmetic progressions. The supplied text gives no resolution, so the conjecture remains open.