Is the following always prime?

Question

For a given $k$ define

$$s_k = 1 + \prod_{i=1}^k p_i$$ $$t_k = \text{NextPrime}(s_k)$$ $$v_k = t_k - s_k +1$$

Where $p_i$ is the $i$th prime number.

Conjecture: $v_k$ is prime

Example: $$k=3$$ $$s_3 = 1+(2)(3)(5)=31$$ $$t_3 = 37 $$ $$v_3 = 37-31 +1 = 7 \,\text{(prime)} $$

Tested numerically up to $k=400$ and it seems to hold.

Answer 1

This is sequence A005235 on OEIS. It is conjectured to always be prime, although it is not known yet. Check the page for useful remarks and references.