= Solution
Let <chemoattractant> reduce the activity per methylated <receptor> by a factor $f$, with $0<f\leq1$, taking the initial factor as one. The active output is $a=fm$. If $V_B$ changes to $V_B'$ but <receptor demethylation> still acts on total methylated <receptor>, the new output is
$$
a_*'=f\frac{K_Br}{V_B'[B]-r}.
$$
Demanding $a_*'=a_* =K_Br/(V_B[B]-r)$ gives
$$
\boxed{V_B'[B]=fV_B[B]+(1-f)r,
\qquad V_B'=fV_B+(1-f)\frac{V_R[R]}{[B]}.}
$$
Reduced <receptor demethylation> lets methylated <receptors> accumulate to $m_*'=m_*/f$, compensating exactly for their reduced activity. During the transition the activity first falls, then rises as <receptor methylation> accumulates. This compensation works only if the required methylated <concentration> is accessible; in particular $m_*/f<E_T$ in a finite <receptor> pool.
The required relation contains both <enzyme> <concentrations>. At fixed rates, increasing <CheR> raises the baseline <receptor methylation> and tumbling output, while increasing <CheB> lowers them. After such a change, the previous fine tuning generally no longer gives <perfect adaptation>. For example the required reverse capacity is $b'=fb+(1-f)r$, so changing $r$ without retuning $b'$ changes the adapted output ratio. It is the \b[precision of adaptation] that is fragile here, in addition to the baseline itself changing. Changing all relevant capacities together in a way preserving the relation is a special exception, not generic robustness.
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