Two-proton decay of the $^{17}$Ne low-lying states populated in the $^1$H($^{18}$Ne,$d$)$^{17}$Ne transfer reaction was studied. The two-proton width $Gamma_{2p}$ of the $^{17}$Ne first excited $3/2^-$ state at $E^*=1.288$ MeV is of importance for the two-proton radioactivity theory and nuclear-astrophysics applications. A dedicated search for the two-proton emission of this state was performed leading to the new upper limit obtained for the width ratio $Gamma_{2p}/Gamma_{gamma} < 1.6(3) times 10^{-4}$. A novel, combined mass method is suggested and tested capable to improve the resolution of the experiment which is a prime significance for the study of nuclear states with extremely small particle-to-gamma width ratios $Gamma_{mathrm{part}}/Gamma_{gamma}$. The condition $Gamma_{mathrm{part}} ll Gamma_{gamma}$ is quite common for the states of astrophysical interest which makes the proposed approach promising in this field.
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