= Solution
Put $p=1-R$, $b=\mu-r\mathbf1$, $A=\sigma\sigma^T$ and $q=R+\varepsilon$. First assume the <covariance matrix> $A$ is invertible and initial <portfolio wealth> is positive. Introduce the accumulated fee process
$$
y_t=aw_t\exp\left(-\frac{\varepsilon}{2}\int_0^t\pi_s^TA\pi_sds\right).
$$
The <Itô formula> and the <self-financing portfolio> equation give
$$
\frac{dy_t}{y_t}=\left(r+b^T\pi_t-\frac{\varepsilon}{2}\pi_t^TA\pi_t\right)dt+\pi_t^T\sigma\,dW_t.
$$
This converts the path-dependent payment into a terminal-utility problem with state $y$. The <volatility-penalized terminal fee> changes its <drift> as well as its <spot volatility>.
Let $J(t,y)$ be the manager's <value function>. Its <Hamilton-Jacobi-Bellman equation>, with terminal condition $J(T,y)=y^p/p$, is
$$
0=J_t+ryJ_y+\sup_\pi\left\{yJ_yb^T\pi+\frac12\left(y^2J_{yy}-\varepsilon yJ_y\right)\pi^TA\pi\right\}.
$$
The <homogeneity> of <CRRA utility> suggests $J(t,y)=f(t)y^p/p$. Since $yJ_y=f(t)y^p$ and $y^2J_{yy}=-Rf(t)y^p$, the portfolio-dependent expression is
$$
f(t)y^p\left(b^T\pi-\frac q2\pi^TA\pi\right).
$$
Its coefficient is positive even when $p<0$. <Completing the square> gives
$$
b^T\pi-\frac q2\pi^TA\pi
=\frac{b^TA^{-1}b}{2q}-\frac q2(\pi-\pi^*)^TA(\pi-\pi^*),
\qquad
\boxed{\pi^*=\frac{A^{-1}(\mu-r\mathbf1)}{R+\varepsilon}}.
$$
Thus the <stock> fractions are constant. Substitution into the <Hamilton-Jacobi-Bellman equation> yields
$$
f(t)=\exp\left[p\left(r+\frac{b^TA^{-1}b}{2q}\right)(T-t)\right],
\qquad
J(0,aw_0)=\frac{(aw_0)^p}{p}\exp\left[p\left(r+\frac{b^TA^{-1}b}{2q}\right)T\right].
$$
In particular, multiplying the fee by $a$ does not affect the optimal <investment portfolio>.
To prove global optimality rather than merely solve a formal <Hamilton-Jacobi-Bellman equation>, write $v_t=\sigma^T\pi_t$, $v^*=\sigma^T\pi^*$ and let $y^*$ be the fee under the proposed constant policy. Then $b^T\pi=qv^*\cdot v$. Define a probability $Q$ by
$$
\frac{dQ}{dP}=\exp\left(pv^*\cdot W_T-\frac12p^2|v^*|^2T\right).
$$
This density is a true <martingale> because $v^*$ is constant. By the <Girsanov theorem>, $W_t^Q=W_t-pv^*t$ is a <Brownian motion> under $Q$. The <geometric Brownian motion> formula for the fee ratio gives
$$
\log\frac{y_T}{y_T^*}
=\int_0^T(v_s-v^*)\cdot dW_s^Q-\frac{1+\varepsilon}{2}\int_0^T|v_s-v^*|^2ds.
$$
For admissible, locally square-integrable exposures with finite accumulated <quadratic variation>, set
$$
Z_T=\exp\left((1+\varepsilon)\int_0^T(v_s-v^*)\cdot dW_s^Q-\frac{(1+\varepsilon)^2}{2}\int_0^T|v_s-v^*|^2ds\right).
$$
This <stochastic exponential> is a positive <local martingale>, so $E_QZ_T\leq1$. Moreover,
$$
\left(\frac{y_T}{y_T^*}\right)^p=Z_T^{p/(1+\varepsilon)},
\qquad
E_P[y_T^p]=E_P[(y_T^*)^p]E_Q[Z_T^{p/(1+\varepsilon)}].
$$
For $p>0$, the exponent $p/(1+\varepsilon)$ lies between zero and one. The <Jensen inequality> for this <concave function> gives $E_Q[Z_T^{p/(1+\varepsilon)}]\leq1$. For $p<0$, the power is <convex> and decreasing, so the <Jensen inequality> gives $E_Q[Z_T^{p/(1+\varepsilon)}]\geq(E_QZ_T)^{p/(1+\varepsilon)}\geq1$; an infinite moment only makes this inequality stronger. Dividing by $p$, with its corresponding sign, proves $E[U(y_T)]\leq E[U(y_T^*)]$ in both cases. Equality is achieved by the constant policy.
For an ordinary terminal-wealth agent with relative risk aversion $q$, the same <Hamilton-Jacobi-Bellman equation> calculation, with no fee penalty, maximizes $b^T\pi-q\pi^TA\pi/2$. It therefore gives exactly the same optimal <investment portfolio>. For $q\ne1$, the preceding <stochastic exponential> and <Jensen inequality> verification applies with penalty zero and exponent $1-q$, proving global optimality for this agent too. \b[The <spot volatility> penalty adds $\varepsilon$ to effective relative risk aversion.] If $q=1$, the printed expression for $U_0$ has a zero denominator; its appropriate limit, after subtracting an irrelevant constant, is $U_0(x)=\log x$, which still gives $\pi^*=A^{-1}b$. In this logarithmic case, the terminal log-wealth difference from the candidate equals $\log Z_T$ for a positive <stochastic exponential> $Z$, so $E[\log Z_T]\leq\log E[Z_T]\leq0$ proves its optimality.
If $A$ is singular, the same conclusion holds in the absence of <arbitrage>, provided $b$ lies in the <image of a linear map> $x\mapsto Ax$: replace $A^{-1}$ by the <Moore-Penrose inverse>, and add any vector in the <kernel> of $A$ to $\pi^*$. These additions change neither returns nor <Brownian portfolio exposures>. If $b$ has a nonzero component in that <kernel>, a zero-volatility position has a nonzero excess return. Scaling its profitable sign gives <arbitrage>, so a finite attained optimum as claimed requires this degeneracy to be excluded.
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