Fix an oracle order $p\ge 2$, a diameter parameter $D>0$, a smoothness parameter $L_p>0$, and an accuracy $\varepsilon>0$. Let $Z\subseteq\mathbb{R}^d$ be a closed convex set with diameter at most $D$:

\[
\sup_{z,z'\in Z}\|z-z'\|_2\le D.
\]

Let $\mathcal{M}_{p}(D,L_p)$ be the class of monotone operators $F:Z\to\mathbb{R}^d$ satisfying

\[
\langle F(z)-F(z'),z-z'\rangle\ge 0
\qquad \forall z,z'\in Z,
\]

and

\[
\|D^{p-1}F(z)-D^{p-1}F(z')\| \le L_p\|z-z'\|_2
\qquad \forall z,z'\in Z.
\]

For $z\in Z$, define the weak gap

\[
\operatorname{Gap}_{F}(z) := \sup_{u\in Z}\langle F(u),z-u\rangle.
\]

A $p$-th order MVI oracle queried at $\bar z\in Z$ returns

\[
F(\bar z),\ DF(\bar z),\ \ldots,\ D^{p-1}F(\bar z).
\]

For $q\in\{1,\ldots,p\}$ and $M>0$, define the full-domain regularized tensor step $A_q(\bar z;F,M)$ to be any point $z^+\in Z$ satisfying

\[
\left\langle
\mathcal{T}_{q-1}^{F}(z^+;\bar z)
+ \frac{M}{q!}\|z^+-\bar z\|_2^{q-1}(z^+-\bar z),
z-z^+
\right\rangle \ge 0
\qquad \forall z\in Z,
\]

where

\[
\mathcal{T}_{q-1}^{F}(z;\bar z)
:= \sum_{k=0}^{q-1} \frac{1}{k!}D^kF(\bar z)[z-\bar z]^k.
\]

Let $\mathcal{A}_{p}^{\mathrm{MVI}}$ be the class of deterministic algorithms that start from $z_0=0$ and, at iteration $t$, choose

\[
\bar z_t\in\operatorname{span}\{z_0,\ldots,z_t\},
\qquad q_t\in\{1,\ldots,p\},
\qquad M_t>0,
\]

then generate

\[
z_{t+1}=A_{q_t}(\bar z_t;F,M_t).
\]

Determine whether there exists a dimension $d=d(T)$, a set $Z\subseteq\mathbb{R}^d$, and an operator $F\in\mathcal{M}_{p}(D,L_p)$ such that every algorithm in $\mathcal{A}_{p}^{\mathrm{MVI}}$ has an output $z_T$ satisfying

\[
\operatorname{Gap}_{F}(z_T) \ge c_p\frac{L_pD^{p+1}}{T^{(p+1)/2}},
\]

where $c_p>0$ depends only on $p$.