Let $0\leq \underline{\sigma} \leq \overline{\sigma}$ be two constant matrices in $\mathbb{S}^d$. Let $W$ be a Brownian motion under the measure $P_0$ and define
$$ \mathcal{P} := \{P^\sigma \colon \sigma \in L^0(F; \mathbb{S}^d) \text{ such that } 0\leq \underline{\sigma} \leq \sigma \leq \overline{\sigma} \} $$
where $P^\sigma = P_0 \circ (X^\sigma)^{-1}$ and $X^{\sigma}_t = \int_0^t \sigma(s) \mathrm{d}W_s$ $\quad$ $P_0$ almost surely and $L^0(F; \mathbb{S}^d)$ denotes the set of F measurable processes with values in $\mathbb{S}^d$. This construction is from "Backward Stochastic Differential Equations" by Jianfeng Zhang, Section 9.2.2.
Is this set $\mathcal{P}$ of non-equivalent martingale measures convex?
EDIT:
Suppose that the probability space is the classical Wiener space, i.e.,
$$ \Omega = \{\omega \in C([0,T],\mathbb{R}^d), \omega_0=0 \} $$
and $P_0$ is the Wiener measure under which $W$ is a Brownian motion.