Asymptotic normality conjecture for sample PLS

Let $\boldsymbol{\beta}_{ls}$ and $\boldsymbol{\beta}_{pls,m}$ denote the population least-squares and population PLS parameters, respectively. Let $\boldsymbol{\beta}_{ls}$ and $\boldsymbol{\beta}_{pls,m}$ be estimated by the sample least-squares estimator $\widehat{\boldsymbol{\beta}}_{ls}$ and sample PLS estimator $\widehat{\boldsymbol{\beta}}_{pls,m}$, both computed from the observed data $(\mathbf{X},\mathbf{y})$. Under Assumptions~ and, suppose that

$$\sqrt{n}(\widehat{\boldsymbol{\beta}}_{ls}-\boldsymbol{\beta}_{ls}) \xrightarrow{d}\mathcal{N}(\mathbf{0}_{p},\sigma_{y}^2\boldsymbol{\Sigma}_{\mathbf{x}}^+)$$

for $n\to\infty$. Let $\mathcal{K}_{m}(\boldsymbol{\Sigma}_{\mathbf{x}},\boldsymbol{\Sigma}_{\mathbf{x},y})$ be the population Krylov space and let $\mathbf{K}_{m}\in\mathbb{R}^{p\times m}$ be any orthonormal basis of it. Define

$$(\boldsymbol{\Sigma}_{\mathbf{x}})_{m}^+=\mathbf{K}_{m}(\mathbf{K}_{m}^{\top}\boldsymbol{\Sigma}_{\mathbf{x}}\mathbf{K}_{m})^{-1}\mathbf{K}_{m}^{\top}.$$

Sample PLS asymptotic normality conjecture. The sample PLS is asymptotically normal with

$$\sqrt{n}(\widehat{\boldsymbol{\beta}}_{pls,m}-\boldsymbol{\beta}_{pls,m}) \xrightarrow[n\to\infty]{d}\mathcal{N}(\mathbf{0}_{p},\sigma_{y}^2(\boldsymbol{\Sigma}_{\mathbf{x}})_{m}^+).$$

This conjecture would extend asymptotic normality from the sample least-squares estimator to sample PLS in the ill-posed linear-regression framework. The supplied text gives no resolution or supporting result beyond stating it as an open conjecture.