The LCV model assumptions are strictly weaker than the assumptions made by MR. Like LCV, a formulation of the MR assumptions is that the bivariate distribution of SNP effect sizes can be expressed in terms of two distributions. In particular, it assumes that the effect size distribution is a mixture of (1') a distribution whose values are proportional for both traits (representing all SNPs that affect the exposure Y1) and (2') a distribution with zero values for the exposure Y1 (representing SNPs that only affect the outcome Y2). These two distributions can be compared with distributions (1) and (2) above. Because (1") is identical to (1) and (2") is a special case of (2), the LCV model assumptions are strictly weaker than the MR assumptions (indeed, much weaker). We also note that the MR model is commonly illustrated with a non-genetic confounder affecting both traits. Our latent variable L is a genetic variable, and it is not analogous to the non-genetic confounder. Similar to MR, LCV is unaffected by nongenetic confounders (such a confounder may result in a phenotypic correlation that is unequal to the genetic correlation).
