= Solution
Take the missing strong-swimmer times $U_j=S_{j1}$, $j=4,\ldots,8$, as the <latent variables>. Their partners' times are $T_j-U_j$. The complete-data <log-likelihood> is
$$
\ell_c=8\log\lambda_1+8\log\lambda_2
-\lambda_1\left(A+\sum_{j=4}^8U_j\right)
-\lambda_2\left(B+\sum_{j=4}^8(T_j-U_j)\right).
$$
For the E-step of the <expectation-maximization algorithm>, use the <conditional split of two exponential times>. At the old rates, with $d_r=\lambda_1^{(r)}-\lambda_2^{(r)}>0$,
$$
f(u\mid T_j=t)=\frac{d_re^{-d_ru}}{1-e^{-d_rt}},\qquad0<u<t.
$$
This is a <truncated exponential distribution>, whose <conditional expectation> is
$$
\boxed{m_j^{(r)}=E[U_j\mid T_j,\theta^{(r)}]
=\frac1{d_r}-\frac{T_j}{e^{d_rT_j}-1}.}
$$
The displayed mean follows by integrating $u e^{-d_ru}$ against the conditional density; the missing weak-swimmer mean is $T_j-m_j^{(r)}$. As $d_r\to0$, the conditional density becomes uniform on $(0,T_j)$ and $m_j^{(r)}\to T_j/2$.
Write $U_r=A+\sum_jm_j^{(r)}$ and $V_r=B+\sum_j(T_j-m_j^{(r)})$. The E-step objective is
$$
Q=8\log\lambda_1+8\log\lambda_2-\lambda_1U_r-\lambda_2V_r.
$$
Setting its <partial derivatives> to zero gives the unconstrained M-step
$$
\boxed{\lambda_1^{(r+1)}=\frac8{U_r},\qquad
\lambda_2^{(r+1)}=\frac8{V_r}.}
$$
These are eight divided by the completed expected totals for the corresponding swimmers; imputing only observed pair totals without the conditional split would not perform the E-step.
The stated ordering must also be enforced. The <ordered-rate exponential M-step> accepts these updates when $U_r<V_r$, so $\lambda_1^{(r+1)}>\lambda_2^{(r+1)}$. Otherwise its maximum on the closed constraint $\lambda_1\geq\lambda_2$ lies at
$$
\boxed{\lambda_1^{(r+1)}=\lambda_2^{(r+1)}
=\frac{16}{U_r+V_r}
=\frac{16}{A+B+\sum_{j=4}^8T_j}.}
$$
Under a strictly open constraint $\lambda_1>\lambda_2$, a boundary optimum is only a supremum; a finite interior maximizer is not assured for every possible sample. One should report that situation rather than swap the labeled swimmer groups. Starting from positive ordered rates, iterate the E-step and constrained M-step until the parameter changes and observed <log-likelihood> improvement are small. <EM likelihood monotonicity> applies to every exact constrained M-step. No numerical swimming times are supplied, so the data-dependent updates, rather than fabricated numerical estimates, are the answer.
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