Lex-plus-powers conjecture for socle dimensions

Let $R=K[x_1,\ldots,x_n]$, and let $I$ be a homogeneous ideal containing a regular sequence of degrees $2\leq a_1\leq\ldots\leq a_n$. Let $L$ be a lex-plus-powers ideal associated with these degrees such that

$$\operatorname{\mathcal{H}}_{R/I}(i)=\operatorname{\mathcal{H}}_{R/L}(i)\quad\text{for all }i\geq0.$$

The socle is the annihilator of the homogeneous maximal ideal in the relevant quotient, and its graded dimensions are measured by the top graded Betti numbers. Lex-plus-powers socle conjecture. The dimension of the socle of $I$ is at most the dimension of the socle of $L$ in every degree; equivalently,

$$\beta_{n,j}(R/L)\geq\beta_{n,j}(R/I)\quad\text{for all }j\geq0.$$

This is the special case of the lex-plus-powers Betti-number conjecture with top homological index $i=n$, and the source states that it is equivalent to the EGH conjecture. The source gives no resolution of the general conjecture.