1. Rewrite the given equation using trigonometric identities:

The given equation is:
$$\left(\frac{3}{4} - \sin^2(\alpha)\right)\left(\frac{3}{4} - \sin^2(3\alpha)\right)\left(\frac{3}{4} - \sin^2(3^2\alpha)\right)\left(\frac{3}{4} - \sin^2(3^3\alpha)\right) = \frac{1}{256}$$

2. **Use the identity for $\sin^2(x)$:**

Recall that $\sin^2(x) = \frac{1 - \cos(2x)}{2}$. Therefore:
$$\frac{3}{4} - \sin^2(x) = \frac{3}{4} - \frac{1 - \cos(2x)}{2} = \frac{3}{4} - \frac{1}{2} + \frac{\cos(2x)}{2} = \frac{1}{4} + \frac{\cos(2x)}{2}$$

3. Simplify each term:

Applying the above identity to each term:
$$\frac{3}{4} - \sin^2(\alpha) = \frac{1}{4} + \frac{\cos(2\alpha)}{2}$$
$$\frac{3}{4} - \sin^2(3\alpha) = \frac{1}{4} + \frac{\cos(6\alpha)}{2}$$
$$\frac{3}{4} - \sin^2(9\alpha) = \frac{1}{4} + \frac{\cos(18\alpha)}{2}$$
$$\frac{3}{4} - \sin^2(27\alpha) = \frac{1}{4} + \frac{\cos(54\alpha)}{2}$$

4. Combine the terms:

The product of these terms is:
$$\left(\frac{1}{4} + \frac{\cos(2\alpha)}{2}\right)\left(\frac{1}{4} + \frac{\cos(6\alpha)}{2}\right)\left(\frac{1}{4} + \frac{\cos(18\alpha)}{2}\right)\left(\frac{1}{4} + \frac{\cos(54\alpha)}{2}\right) = \frac{1}{256}$$

5. Simplify the equation:

Notice that $\frac{1}{256} = \left(\frac{1}{4}\right)^4$. Therefore, each term must be equal to $\frac{1}{4}$:
$$\frac{1}{4} + \frac{\cos(2\alpha)}{2} = \frac{1}{4}$$
$$\frac{1}{4} + \frac{\cos(6\alpha)}{2} = \frac{1}{4}$$
$$\frac{1}{4} + \frac{\cos(18\alpha)}{2} = \frac{1}{4}$$
$$\frac{1}{4} + \frac{\cos(54\alpha)}{2} = \frac{1}{4}$$

6. **Solve for $\alpha$:**

Each equation simplifies to:
$$\frac{\cos(2\alpha)}{2} = 0 \implies \cos(2\alpha) = 0 \implies 2\alpha = 90^\circ \implies \alpha = 45^\circ$$

However, we need to find the least positive angle $\alpha$ such that:
$$\sin(81\alpha) = \sin(\alpha)$$

This implies:
$$81\alpha = 180^\circ k + \alpha \quad \text{or} \quad 81\alpha = 180^\circ k - \alpha$$

For the smallest positive $\alpha$:
$$81\alpha = 180^\circ + \alpha \implies 80\alpha = 180^\circ \implies \alpha = \frac{180^\circ}{80} = \frac{9^\circ}{4}$$

7. **Express $\alpha$ in simplest form:**

$\alpha = \frac{9}{4}^\circ$. In degrees, this is $\frac{9}{4}$.

8. **Find $m + n$:**

Here, $m = 9$ and $n = 4$. Therefore, $m + n = 9 + 4 = 13$.

The final answer is $\boxed{13}$.