Diamantis–Rolen odd-part containment conjecture for derivative period polynomials

Let $k$ be an even weight, let $f\in S_k(\mathrm{SL}_2(\mathbb Z))$ be a normalized Hecke eigenform, and let $m\geq 0$ be an integer. Define the derivative period polynomial $Q_{f,m}$ and its odd part $Q^-_{f,m}$ by

$$Q_{f,m}(z)=\sum_{j=0}^{k-2}\binom{k-2}{j}i^{1-j}\Lambda_f^{(m)}(j+1)z^{k-2-j},$$ $$Q^-_{f,m}(z)=\frac{Q_{f,m}(z)-Q_{f,m}(-z)}{2}.$$

Let $\mathbb T=\{z\in\mathbb C:|z|=1\}$ and let $\mathcal Z(P)$ denote the set of distinct zeros of a polynomial $P$.

Diamantis–Rolen odd-part containment conjecture. There is a real number $b=b(f,m,k)>0$ such that

$$\mathcal Z(Q^-_{f,m})\subseteq\mathbb T\cup\{0,\pm b,\pm b^{-1}\}.$$

The exceptional real reciprocal orbit may be absent.

This predicts that, apart from the forced zero at the origin, every zero of the odd part of a derivative period polynomial lies on the unit circle except possibly for a real reciprocal orbit. The statement extends the known rigid zero geometry for the odd period polynomial when $m=0$; its validity for all even weights, normalized Hecke eigenforms, and $m\geq0$ remains open.