An effective prime sequence?

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive integers, where $a_1 = 2$ and each subsequent term can be expressed as 1 plus the product of all its previous terms. For example, $a_2 = 1+a_1= 3$ and $a_3 = 1+a_1 \times a_2=1 + 2\times3=7.$

Note that $a_1, a_2,a_3$ are all prime numbers.

Is it true that $a_4, a_5, a_6,\ldots$ are all prime numbers as well?