The spider lower-bound equality conjecture for maximum rectilinear crossing numbers

Let $S$ be a spider with $k\geq 3$ legs, whose lengths satisfy

$$\ell_1\geq \ell_2\geq \cdots \geq \ell_k.$$

Let $\vartheta(S)$ denote the thrackle bound, and let $\max\text{-}\overline{\operatorname{cr}}(S)$ denote the maximum rectilinear crossing number of $S$.

Spider lower-bound conjecture.

$$\max\text{-}\overline{\operatorname{cr}}(S)=\vartheta(S)-\sum_{i=3}^k(\ell_i-1)\left\lfloor\frac{i-1}{2}\right\rfloor.$$

The formula asserts that the lower bound obtained from the paper's drawing algorithm is exact for every spider. The authors motivate it by noting that the upper and lower bounds agree when all legs have length two, while their method does not otherwise generalize; the supplied text gives no resolution of the general conjecture.