The lower bound for binary ones in triangular numbers

Let $t_n=n(n+1)/2$ be the $n$th triangular number, and let $(n)_2^1$ denote the positions of the $1$s in the binary representation of $n$.

Binary-weight conjecture. If $n\in\mathbb{N}$ is such that $|(n)_2^1|\geq 6$, then $|(t_n)_2^1|\geq 4$.

This conjecture concerns how the number of $1$s in the binary representation behaves under the triangular-number map. The supplied text gives no resolution, so its status remains open.