Shumakovitch–Turner recurrence conjecture for torus-knot Khovanov polynomials

Let $K_n(t,q)$ denote the Poincaré polynomial of the Khovanov homology of the torus knot $T(n+1,n)$. Shumakovitch–Turner recurrence conjecture. The polynomials satisfy

$$K_0=K_1=q^{-1}+q,\qquad K_2=q+q^3+t^2q^5+t^3q^9,$$

and, for $n\ge3$,

$$K_n=q^{2n-2}K_{n-1}+t^{2n-2}q^{6n-6}K_{n-2}+t^{2n-1}q^{8n-8}K_{n-3}.$$

The recurrence is stated as a conjecture for the Poincaré polynomials of the Khovanov homology of the family $T(n+1,n)$; the supplied text gives no resolution status.