Wilkie's well known question asks whether $I\Delta_{0}$ proves the unboundedness of primes. We know that by adding a sentence to $I\Delta_{0}$ which says "the exponential function is total", it is possible to prove the unboundedness of primes. This sentence is $\Pi_{2}$. Suppose $\Pi_{1}\text{-Th}(\mathbb{N})$ denotes the set of all $\Pi_{1}$ sentences that are true in $\mathbb{N}$. My question is:

Is it known that $I\Delta_{0} +\Pi_{1}\text{-Th}(\mathbb{N})$ proves the unboundedness of primes?