Model
Consider a single-agent decision problem with uncertainty.
A decision maker (DM) has to choose action $y\in \mathcal{Y}$ possibly without being fully aware of the state of the world. $\mathcal{Y}$ is a finite set. The state of the world is a random variable $V$ with support $\mathcal{V}$. When the DM chooses action $y\in \mathcal{Y}$ and the state of the world is $v\in \mathcal{V}$, she receives the payoff $u(y,v)$. Let $P_V\in \Delta(\mathcal{V})$ be the DM's prior about the state of the world $V$.
The DM also processes some signal $T$ with support $\mathcal{T}$ and distribution $P_{T|V}$ conditional on $V$ to refine his prior and get a posterior on $V$, denoted by $P_{V|T}$, via the Bayes rule.
A strategy for the DM is a distribution of actions conditional on the signal, which we denote by $P_{Y|T}$. Such a strategy is optimal if it maximises his expected payoff, where the expectation is computed using the posterior $P_{V|T}$.
Hereafter, we call $S\equiv (\mathcal{T}, P_{T|V})$ as the DM's information structure.
Question
In the worst-case scenario, the signal is uninformative about $V$ (null information structure). In this scenario, the DM with assigned state $v$ will choose based on the prior $P_V$ and get the utility
$$
\bar{u}(v)\equiv u\Big(\text{argmax}_{y\in \mathcal{Y}} \int_\mathcal{V} u(y,x) dP_V(x), v\Big).
$$
Can we show that $\bar{u}(v)$ is the lowest utility that the DM can attain across every possible information structures? In other words, take any information structure that is at least as informative as the null information structure; suppose the DM gets some signal $t$ from such information structure; does it hold that
$$
u\Big(\text{argmax}_{y\in \mathcal{Y}} \int_\mathcal{V} u(y,x) dP_{V|T}(x|t), v\Big)\geq \bar{u}(v)\quad ?
$$