Rosen's lifting conjecture for multiple harmonic sums

Let $k$ be a positive integer and let $p_1$ be a prime greater than $k+1$. For each index $\boldsymbol{k}$, let $H_{p-1}(\boldsymbol{k})$ be the multiple harmonic sum, and let $c_{\boldsymbol{k}}$ be a finite set of rational coefficients indexed by indices of weight $k$. Suppose

$$\sum_{\operatorname{wt}(\boldsymbol{k})=k}c_{\boldsymbol{k}}H_{p-1}(\boldsymbol{k})\equiv0\pmod p$$

for all primes $p\ge p_1$. Rosen's lifting conjecture. For each $n\ge2$, there exists a prime $p_n$ and a finite set of rational numbers $c_{\boldsymbol{k}}^{(l)}$, indexed by $1\le l\le n-1$ and indices of weight $k+l$, such that $p_n\ge p_{n-1}$ and

$$\sum_{\operatorname{wt}(\boldsymbol{k})=k}c_{\boldsymbol{k}}H_{p-1}(\boldsymbol{k})+\sum_{l=1}^{n-1}p^l\sum_{\operatorname{wt}(\boldsymbol{k})=k+l}c_{\boldsymbol{k}}^{(l)}H_{p-1}(\boldsymbol{k})\equiv0\pmod {p^n}$$

for all primes $p\ge p_n$. This is a lifting principle: a congruence modulo $p$ should admit coefficient corrections that lift it to every higher power of $p$. The supplied text presents it as a version of Rosen's lifting conjecture and gives no resolution status.