Convert a Park McClellan FIR Solution to Achieve Stop-band Roll-off
That trick must have to do with manipulating the first and the last tap of the Parks-McClellan (Chebyshev) solution. This is also explained in the last paragraph of this answer, which deals with a very extreme case. For equi-ripple designs the first and the last tap tend to be a bit larger in magnitude than what would be expected if one just extrapolated the impulse response. These two extremal taps correspond to time-domain echos and generate the equi-ripple behavior in the frequency domain. By decreasing the size of these two taps, the stop band ripple can be reduced. Any heuristic can be used to do so. I don't know what Harris suggested, but by simply choosing $h[0]=h[1]/2$ and $h[N-1]=h[N-2]/2$ (where $N$ is the filter length) I get an even faster roll-off (albeit a less regular one), at apparently no extra cost (compared to the solution in the question):





Note that this trick only works well for filters with a relatively narrow pass band. Equi-ripple filters with wider pass bands usually do not show that irregularity in their impulse response.