The 2-adic valuation conjecture for even-colored 2-ary partition functions

Let $c_m(n)$ denote the $m$-colored binary partition function, and let $u_2$ be the 2-adic valuation. For positive integers $k,n$ and nonnegative integers $k$ as specified below, consider the differences of the values of $c_m$. 2-adic valuation conjecture. For $kin\mathbb{N}_{+}$ and each $n\in\mathbb{N}_{+}$,

$$\nu_{2}(c_{2k}(4n)-c_{2k}(n))=\nu_{2}(n)+2\nu_{2}(k)+3.$$

Moreover, for $k\in\mathbb{N}$ and $n\in\mathbb{N}_{+}$,

$$\nu_{2}(c_{4k+1}(4n)-c_{4k+1}(n))\geq\nu_{2}(n)+3,$$ $$\nu_{2}(c_{4k+3}(4n)-c_{4k+3}(n))\geq\nu_{2}(n)+6.$$

In each case, equality holds for infinitely many $n\in\mathbb{N}$. These predictions are motivated by known sharp congruences for the ordinary binary partition function and numerical computations for $m\leq10$ and $n\leq10^5$; their general validity remains open.