Dąbrowski's conjecture on smooth numbers of the form x^2-1

Let $p_1=2,p_2=3,p_3=5,\ldots,p_k$ be the first $k$ primes. For positive integers $x$ and nonnegative integers $\alpha_1,\ldots,\alpha_k$, consider the Diophantine equation

$$x^2-1=p_1^{\alpha_1}\cdots p_k^{\alpha_k}.$$

Dąbrowski's conjecture. The equation has exactly $28$ solutions $(x;\alpha_1,\ldots,\alpha_k)$ in positive integers, namely the solutions listed in the statement: $(3;3)$; $(5;3,1),(7;4,1),(17;5,2)$; $(11;3,1,1),(19;3,2,1),(31;6,1,1),(49;5,1,2),(161;6,4,1)$; $(29;3,1,1,1),(41;4,1,1,1),(71;4,2,1,1),(251;3,2,3,1),(449;7,2,2,1),(4801;7,1,2,4),(8749;3,7,4,1)$; $(769;9,1,1,1,1),(881;5,2,1,2,1),(1079;4,3,1,2,1),(6049;6,3,2,1,2),(19601;5,4,2,2,2)$; $(3431;4,1,1,3,1,1),(4159;7,3,1,1,1,1),(246401;8,6,2,1,1,2)$; $(1429;3,1,1,1,1,1,1),(24751;5,2,3,1,1,1,1),(388961;6,4,1,4,1,1,1)$; and $(1267111;4,3,1,1,3,1,1,2)$.

This conjecture asks for a complete determination of the solutions whose value $x^2-1$ has no prime factors beyond the first $k$ primes. The source attributes the formulation to Dąbrowski; the provided text gives no evidence that the conjecture has been proved or disproved.