Very efficient format conjecture for moment matrix extension

Let $n$ and $r$ be non-negative integers, and let $E_1$ and $Y$ denote the edge and monomial-index sets used in the moment matrix extension algorithm. For an integer $c$, write

$$R_c(n)=\sum_{j=0}^c(n-j+1).$$

A format is called very efficient when the corresponding linear system has the required tall-matrix property, and efficient when the algorithm decomposes generic tensors in that format.

Very efficient format conjecture.

1. The format $(n,R_c(n))$ is very efficient if $c$ is such that $|E_1|\geq |Y|$.
2. The format $(n,r)$ is efficient if there exists $c$ with $r\leq R_c(n)$ such that $(n,R_c(n))$ is very efficient.

This conjecture would extend the proved efficient-format range from ranks of order $n$ to ranks of order $n^2$, making moment matrix extension and tensor decomposition effective for a substantially larger family of tensors. The paper reports a computer-assisted verification of the conjectured behavior up to $n=17$.