The torus-knot slope-jump conjecture for the invariant $\Phi$

Let $T_{n,n+1}$ denote the torus knot with parameters $n$ and $n+1$, let $\Phi_{T_{n,n+1}}$ be the associated invariant, and write $\Delta \Phi'_{T_{n,n+1}}(\alpha)$ for the jump in the slope of its function at $\alpha$. Let $s(T_{n,n+1})$ denote the concordance invariant appearing in the statement. For other torus knots, use the notation $\Phi_{T_{p,q}}$ for the corresponding invariant.

Torus-knot slope-jump conjecture. For the torus knot $T_{n,n+1}$, the jump in the slope $\Delta \Phi'_{T_{n,n+1}}(\alpha)$ is equal to $0$ except when $\alpha=\frac{2+2k}{2+3k}$ for $0\leq k<n$ and $k$ has the same parity as $n$, when it is $-4-6k$. Together with

$$s(T_{n,n+1})=n(n-1),$$

this determines $\Phi_{T_{n,n-1}}$. For other torus knots, $\Phi$ is determined by the recurrence

$$\Phi_{T_{p,q+p}}=\Phi_{T_{p,q}}+\Phi_{T_{p,p+1}}.$$

The pattern is motivated by computations for small torus knots and would give the invariant $\Phi$ on the family of torus knots. The supplied text does not provide a resolution or further evidence beyond describing the pattern as a conjecture.