Try adding valid constraints $$ y_{i,j} \le \sum_{(i,k): k \in \tilde{V} \setminus \{j\}} y_{i,k} \quad\text{for $(i,j)$ such that $\hat{y}_i = 0$ and $j \in \tilde{V}$} $$ that enforce the logical implications $$(y_{i,j} \land \lnot\hat{y}_i \land [j \in \tilde{V}]) \implies \bigvee_{(i,k): k \in \tilde{V} \setminus \{j\}} y_{i,k}$$

Try adding valid constraints $$ y_{i,j} \le \sum_{(i,k): k \in \tilde{V} \setminus \{j\}} y_{i,k} \quad\text{for $(i,j)$ such that $\hat{y}_i = 0$ and $j \in \tilde{V}\setminus\{i\}$} $$ that enforce the logical implications $$(y_{i,j} \land \lnot\hat{y}_i \land [j \in \tilde{V} \setminus\{i\}]) \implies \bigvee_{(i,k): k \in \tilde{V} \setminus \{j\}} y_{i,k}$$

Try adding valid constraints $$ y_{i,j} \le \sum_{(i,k): k \in \tilde{V} \setminus \{j\}} y_{i,k} \quad\text{for $(i,j)$ such that $\tilde{y}_i = 0$ and $j \in \tilde{V}$} $$ that enforce the logical implications $$(y_{i,j} \land \lnot\tilde{y}_i \land [j \in \tilde{V}]) \implies \bigvee_{(i,k): k \in \tilde{V} \setminus \{j\}} y_{i,k}$$

Try adding valid constraints $$ y_{i,j} \le \sum_{(i,k): k \in \tilde{V} \setminus \{j\}} y_{i,k} \quad\text{for $(i,j)$ such that $\hat{y}_i = 0$ and $j \in \tilde{V}$} $$ that enforce the logical implications $$(y_{i,j} \land \lnot\hat{y}_i \land [j \in \tilde{V}]) \implies \bigvee_{(i,k): k \in \tilde{V} \setminus \{j\}} y_{i,k}$$