Integral Rasmussen invariant conjecture for Whitehead doubles of odd torus knots

Let $T(a,b)$ denote the $(a,b)$-torus knot, let $W_+(J,t)$ denote the positive $t$-twisted Whitehead double of a knot $J$, and let $\overline{K}$ denote the mirror of $K$. For a knot, $s^{\mathbb{Z}}(K)$ is the integral Rasmussen invariant recorded as the relevant ordered pair.

Whitehead-double torus-knot conjecture. If $n$ is a positive integer and $K$ is the mirror of

$$W_+\bigl(T(2,2n+1),3n\bigr),$$

then

$$s^{\mathbb{Z}}(K)=(0,2^n).$$

The conjecture is motivated by computed Whitehead-double examples and is closely related to a conjecture that the flat $2$-cabling of $T(2,2n+1)$ contains Khovanov-homology torsion of order $2^n$. Computations establish the relevant torsion in low cases, but its survival to the $E_\infty$ page, needed for the asserted invariant formula, remains to be checked.