filters - Matlab `filtfilt` provides excessive transient

I observed that filtfilt suffers from an undesired behavior when I provide IIR bandpass filters having steep transition bands. Specifically, the output signal exhibits excessive transient response; nevertheless, this behaviour does not emerge if I use a 'home-made' version of zero phase filtering based on filter. I guess that it is not a numerical issue involving the filter coefficients or structure, nor the filter function.

% design bandpass filter having transition bandwidth of 200 Hz (Fs = 8000)
bp = designfilt('bandpassiir', 'StopbandFrequency1', 50, 'PassbandFrequency1', 250,...
    'PassbandFrequency2', 3600, 'StopbandFrequency2', 3800, 'StopbandAttenuation1', 30,...

    'PassbandRipple', 0.1, 'StopbandAttenuation2', 30, 'SampleRate', 8000, 'DesignMethod', 'cheby2');
% check stability
assert(isstable(bp),'Unstable filter');
% apply filtfilt to a random (white) long input signal; output signal shows an undesirable transient
x = randn(2^20,1);
y = filtfilt(bp,x);
% apply 'home-made' filtfilt to the same input; output signal shows a more acceptable transient
y2 = flipud(filter(bp,flipud(filter(bp,x))));
% compare effects
figure; semilogy(abs(y-y2));

As a rule of thumb, the effect grows as the transition bands get narrower, while it tends to vanish as they get broader.

Where is the problem? Have I missed some recommendations or hints in the function's help?