Equality of real and complex Markov-Bernstein constants

Let ${\boldsymbol{h}} \in {\mathbb C}_{++}^n$, and define

$$M_{\, \ell}({\boldsymbol{h}}) = \max_{\|p\|_{\infty} \le 1,\ p \in {\cal P}_{{\boldsymbol{h}}}} \bigl\|p^{\,(\ell)}\bigr\|_{\infty},$$

with

$$M_{\, \ell,n}=\max_{{\boldsymbol{h}}\in{\cal D}_n}M_{\,\ell}({\boldsymbol{h}}).$$

For a real positive vector ${\boldsymbol{h}}\in{\cal I}_n=(0,1]^n$, define

$$m_{\,\ell}({\boldsymbol{h}})=\max_{\|p\|_{\infty}\le 1,\ p\in{\cal RP}_{{\boldsymbol{h}}}}\bigl\|p^{\,(\ell)}\bigr\|_{\infty},$$

with

$$m_{\,\ell,n}=\max_{{\boldsymbol{h}}\in{\cal I}_n}m_{\,\ell}({\boldsymbol{h}}).$$

The equality conjecture. For all $\ell,n\in{\mathbb N}$, we have

$$m_{\,\ell,n}=M_{\,\ell,n}.$$

This asserts that allowing complex exponential parameters does not increase the sharp $\ell$-th derivative Markov-Bernstein constant beyond the value attained by real exponential polynomials. The source presents this as its main conjecture; no resolution is supplied here.