Conjecture on the third term of lens surgery polynomials

Let $p$ and $k$ be relatively prime positive integers. Let $K_{p,k}$ be the lens space knot in the homology sphere $Y_{p,k}$ dual to a simple $(1,1)$-knot in a lens space, with parameters $(p,k)$. Let $\Delta_{K_{p,k}}(t)$ denote its symmetrized Alexander polynomial, and suppose its third top term is non-zero. Third-term conjecture. Then there is an integer $g$ such that

$$\Delta_{K_{p,k}}(t)=\Delta_{T(2,2g+1)}(t),$$

or, equivalently, $k=2$. The conjecture concerns a criterion forcing the Alexander polynomial of these lens space knots to be that of an odd $(2,2g+1)$-torus knot; the supplied text does not state whether it has been resolved.