It is true that is a measurable degradation, but it is at a level that is of academic interest only. Below is the THD+N graph of the headphone output. When the graph is a straight line that slopes down by 10 dB every 10x increase in output, it is an indication that the THD+N is totally dominated by a constant level idle noise. And this is the case for the DX3 Pro+ in pretty much its entire operating range.
To calculate the voltage of this idle noise, I took the point on the (high gain) curve at 10 μW which gave a THD+N of -77 dB. The load is 300 Ω and the output voltage at 10 μW is sqrt(1e-5*300) = 0.0548 Vrms. The idle noise level is therefore 0.0548*(10^(-77/20)) = 7.7 μV.
Relative to the standard voltage of 2.83 V speaker sensitivities are reported, with the 19.1 dB gain of the PA5, 7.7 μV is 20*log10(7.7e-6/2.82) + 19.1 = -92 dB. Therefore, if your speaker has a sensitivity of 90 dB SPL at 2.83 V, the noise contributed by the DX3 Pro+ headphone out will be -2 dB SPL at 1 m, which is completely insignificant.
Amir's test of the RCA output reported a dynamic range of 118 dB. Whilst it is mathematically better than the headphone output of 111 dB (= -92 - 19, when not counting the PA5 gain of 19 dB), it isn't anything close to a practical benefit.