A 3-adic valuation formula for cubic binomial sums

For an integer $n\geq 0$, define

$$T_n=\sum_{k=0}^n\binom{n}{k}^3 2^k.$$

Let $s_3(m)$ denote the sum of the base-$3$ digits of the nonnegative integer $m$. The cubic binomial-sum valuation conjecture. For every integer $n\geq 0$,

$$\nu_3(T_n)=\begin{cases} s_3\left(\frac{n-1}{2}\right)+1,&\text{if }n\equiv -1\pmod 6;\\ s_3\left(\left\lfloor\frac{n+1}{2}\right\rfloor\right),&\text{otherwise.} \end{cases}$$

The source poses this as an open question along with results on related 3-adic valuations. No resolution is supplied in the given text.