Probabilistic representer conjecture under integrability of feature maps

Let $X$ be the underlying space, let $\mathbb{F}$ be the scalar field, let $\mu$ be a probability measure on $X$, and let $\phi:X\to\mathscr{H}(K)$ be the feature map associated with the reproducing kernel Hilbert space $\mathscr{H}(K)$. Let $\hat{f}$ be the unique optimizer in Theorem. Assume that, for every $\mathbb{F}$-measure $\xi$ on $X$ whose support is contained in the support of $\mu$, one has

$$\int_X \|\phi(x)\|_{\mathscr{H}(K)}\,d|\xi|(x)<\infty.$$

Probabilistic representer conjecture. There exists an $\mathbb{F}$-measure $\nu$ on $X$, with support contained in the support of $\mu$, such that

$$\hat{f}=\int_X \phi(x)\,d\nu(x).$$

This conjecture asserts that the optimizer has a direct measure representation under the stated integrability condition, rather than only an approximation by a sequence of measures. It is presented as a conjectural strengthening of the probabilistic representer theorem, and no resolution is given in the source.