Non-simple telltale intersection-multiplicity conjecture

Let $R\to\mathbb P^1$ be a rational elliptic surface, let $L$ determine a Severi curve with normalization $\widetilde{V}(L)$, and let $f_L:\widetilde{V}(L)\to\operatorname{Sym}^2(\mathbb P^1)$ record the two branch points of a rational bisection. Let $\Delta\subset\operatorname{Sym}^2(\mathbb P^1)$ be the diagonal conic, and let $s_1+s_2+mN$ be a non-simple telltale, where $s_1,s_2$ are sections, $m$ is a positive integer, and $N$ is a nodal fiber. Write $\sigma_1(m)$ for the sum of the positive divisors of $m$. Non-simple telltale multiplicity conjecture. The intersection multiplicity of $f_L$ with $\Delta$ at $s_1+s_2+mN$ is

$$\begin{cases} 2\sigma_1(m), & s_1\ne s_2,\\ \sigma_1(m), & s_1=s_2,\ m\text{ is not a square},\\ \sigma_1(m)-1, & s_1=s_2,\ m\text{ is a square}. \end{cases}$$

Together with the transverse contribution from simple telltales, this conjecture would control the intersection number governing the degree calculation for the Severi curves. Its status is not resolved in the supplied material.