Consider the so-called Stern's triangle (refer to these slides by R. Stanley), we denote here by $a_n(k)$. In an article Some linear recurrences motivated by Stern’s diatomic array, Stanley provided the generating functional format $$\sum_{k\geq0}a_n(k)x^k=\prod_{i=0}^{n-1}(1+x^{2^i}+x^{2\cdot2^i}).$$ The author also studied power sums, such as, $$\pmb{v}_{n,r}:=\sum_{k\geq0}a_n(k)^r.$$ At present, starting with the recurrences satisfied by Stern's sequence, $a_n(k)$; that is, $a_n(2k+1)=a_{n-1}(k)$ and $a_n(2k)=a_{n-1}(k)+a_{n-1}(k+1)$, one may compute a system of recursive equations for $\pmb{v}_{n,r}$. This leads to some matrices (my interest is in their determinants). An example may be found in Proof of a conjecture of Stanley about Stern's array (page 2), a follow up paper by D. Speyer. For another example look back into the 2nd paper mentioned above (page 6).
So, I would like to ask:
QUESTION. Suppose $M(r)$ is the matrix, of size $(r+1)\times(r+1)$, of entries $$M_{i,j}(r)=\binom{i}j+\binom{2r-i}{j-i}+\binom{2r-i}j-t_{i,j} \qquad 0\leq i,j\leq r$$ where $t_{i,j}=0$ except for the last column $t_{i,r}=\binom{2r-i}r$. Is the following true? $$\det M(r)=(-1)^{\delta_r}\,\,2^{\lfloor \frac{r+3}3\rfloor}$$ with $\delta_r$ a period-$12$ sequence in the pattern $[0,1,0,1,1,1,1,0,1,0,0,0]$. Here $\lfloor\cdot\rfloor$ is the floor function.