To make the partial transpose a complete positive and therefore physical map, one has to mix it with enough of the maximally mixed state to offset the negative eigenvalues.
The most negative eigenvalue is obtained when partial transpose is applied on the maximally entangled state which is $-\frac{1}{2}$.
Therefore, $$\widetilde{I \otimes \Lambda} = (1-p)(I \otimes \Lambda) \rho + p\frac{I \otimes I}{4}$$ where $p = \frac{2}{3}$ will sufficiently offset the values. However, in this paper by Horodecki and Ekert, they say $p$ needs to be greater than $\frac{8}{9}$, which I can't understand why.