Rational-center valuation conjecture for the Tribonacci sequence

Let $p$ be a prime number, and let $T(n)$ be the Tribonacci sequence defined by $T(0)=0$, $T(1)=T(2)=1$, and $T(n+3)=T(n+2)+T(n+1)+T(n)$. Write $\nu_p(m)$ for the exponent of $p$ in a nonzero integer $m$, with $\nu_p(0)=\infty$. Rational-center valuation conjecture. There exists a positive integer $Q$ such that, for every $i\in\{0,1,\ldots,Q-1\}$, one of the following holds: (C) there exists $\kappa_i\in\mathbb Z_{\ge0}$ such that, for all but finitely many integers $n\equiv i\pmod Q$, $\nu_p(T(n))=\kappa_i$; or (L) there exist $a_i\in\mathbb Q$, $\kappa_i\in\mathbb Z$, and $\mu_i\in\mathbb Z_{>0}$ satisfying $\nu_p(a_i-i)\ge\nu_p(Q)$ such that, for all but finitely many integers $n\equiv i\pmod Q$, $\nu_p(T(n))=\kappa_i+\mu_i\nu_p(n-a_i)$. This is proposed as a possible repair of the preceding conjecture after its failure; no resolution is supplied in the text.