= Solution
Put $A=\sum_{i=1}^3S_{i1}$, $B=\sum_{i=1}^3S_{i2}$ and $d=\lambda_1-\lambda_2$. <Independence> of the pairs makes their likelihood contributions multiply. The three fully observed pairs contribute $(\lambda_1\lambda_2)^3e^{-\lambda_1A-\lambda_2B}$; each remaining pair contributes its <hypoexponential distribution> density. Thus the observed-data <log-likelihood> is
$$
\boxed{\ell(\lambda_1,\lambda_2)=8\log\lambda_1+8\log\lambda_2
-\lambda_1A-\lambda_2B-5\log d
+\sum_{j=4}^8\log(e^{-\lambda_2T_j}-e^{-\lambda_1T_j}).}
$$
An equivalent expression, convenient for calculations, is
$$
\ell=8\log\lambda_1+8\log\lambda_2-5\log d
-\lambda_1A-\lambda_2\left(B+\sum_{j=4}^8T_j\right)
+\sum_{j=4}^8\log(1-e^{-dT_j}).
$$
The apparent singularity as $d\downarrow0$ cancels between the last sum and $-5\log d$. The resulting equal-rate limit is finite for positive observed times.
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