Let $s(x_1,x_2) \in \mathbb C [[x_1, x_2]] [x_1^{-1},x_2^{-1}]$ with all coefficients equal to one or zero. For example $$ s(x_1,x_2)=\frac1{(1-x_1)(1-x_2)}$$ Now let $$ t(x_1,x_2) = \sum_{i=1}^n s(x_1^{a_i} x_2^{b_i},x_1^{c_i} x_2^{d_i})$$ where $a_i,b_i,c_i,d_i \in \mathbb Z$ for all $i$, and $a_i b_i \neq 0 \neq c_i d_i$. For example $$t(x_1,x_2)=\frac1{(1-x_1)(1-x_2)} +\frac1{(1-x_1^{-1})(1-x_1^{-1} x_2)}$$
Is it true that if $\lim_{x_1,x_2\to1} t(x_1,x_2)$ is finite, then $t(x_1,x_2) \in \mathbb C [x_1^\pm, x_2^\pm]$?