Skolem's local-global conjecture for purely exponential Diophantine equations

Consider the purely exponential Diophantine equation

$$c_1a_{11}^{x_{11}}\cdots a_{1l_1}^{x_{1l_1}}+c_2a_{21}^{x_{21}}\cdots a_{2l_2}^{x_{2l_2}}+\cdots+c_ka_{k1}^{x_{k1}}\cdots a_{kl_k}^{x_{kl_k}}=0,$$

where the variables $x_{ij}$ are positive integers. Skolem's conjecture. If this equation has no solution, then there exists an integer $M\geq 2$ such that the congruence

$$c_1a_{11}^{x_{11}}\cdots a_{1l_1}^{x_{1l_1}}+c_2a_{21}^{x_{21}}\cdots a_{2l_2}^{x_{2l_2}}+\cdots+c_ka_{k1}^{x_{k1}}\cdots a_{kl_k}^{x_{kl_k}}\equiv 0\pmod{M}$$

does not hold for any positive integers $x_{ij}$ with $i=1,2,\dots,k$ and $j=1,2,\dots,l_k$. This is a local-global principle: an absence of integral solutions should already be detected modulo a suitable modulus, but the general conjecture remains open; the paper presents the statement as an application to the displayed equation.