Non-institutionalized wife wants to sell home. Can she keep 50% of equity realized from the sale, or does entire equity go the state?

Estate Recovery Act requires equity on couple's home be used to pay SNF after death of 2nd spouse. In the meantime, 2nd spouse wants to sell in a J/T/R/O/S - joint tenancy right-of-survivorship state. Typically property can be sold and laws acknowledge each spouse owns 50% equity which they can invest together or invest as individuals.
Medicaid says spouse's SHARE OF assets will not count toward ELIGIBILITY for Medicaid. So spouse qualifies for SNF. Non-institutionalized spouse wants to keep 50% proceeds of the couple's equity as an individual asset. Put another way, we don't know if 100% of the sales proceeds must be paid to the state.
Estate Recovery outcomes for the spouse's 50% share realized WITHIN the surviving spouse's lifetime just isn't evident.