A conjecture of Radu and Sellers on congruences modulo powers of 2 for broken 3-diamond partitions

In 2007, Andrews and Paule introduced the family of functions ${\mathrm{\Delta }}_k(n)$, which enumerate the number of broken $k$-diamond partitions for a fixed positive integer $k$. In 2013, Radu and Sellers completely characterized the parity of ${\mathrm{\Delta }}_3(8n+r)$ for certain values of $r$ and proposed a conjecture on congruences modulo powers of $2$ for broken $3$-diamond partitions. In this paper, we employ an unconventional $U$-sequence to resolve the revised conjecture put forward by Radu and Sellers.