Lifting and overlapping conjecture for minimal Markov bases of the p1 models

Let $I_{\mathcal A_n}$ be the toric ideal associated with the base $p_1$ model on $n$ nodes. The operations of lifting and overlapping construct binomials for larger subnetworks from binomials on smaller subnetworks. A minimal Markov or Gröbner basis is a minimal generating set of the corresponding toric ideal, and an $(n-1)$-node subnetwork is obtained by deleting one node.

Lifting and overlapping conjecture. Minimal Markov and Gröbner bases for the $p_1$ models can be obtained from Markov and Gröbner bases of $I_{\mathcal A_n}$ by repeated lifting and overlapping of the binomials in the minimal Markov bases of various $(n-1)$-node subnetworks.

This conjecture proposes a recursive construction of minimal bases from smaller subnetworks; the source gives no resolution and treats it as an open direction.