A much more accurate account is in THE CHARGE OF THE ELECTRON (1935):
In 1929 Prof. R. T. Birge (reference 1) published an acutely critical and masterly survey of our knowledge of the fundamental physical constants. It was a very timely summary, and it undoubtedly-if we may borrow from the vocabulary of
another trade-did much to promote a desirable "constants-consciousness ” in the
general body of physicists. This especially applies to two of the atomic constants,
the charge e and the specific charge e/m of the electron.
At the time of Birge’s report, there were two distinct, and apparently equally
well authenticated, values of e/m. Direct deflection methods gave 1.769 x 10^7,
while spectroscopic methods gave 1.761 x 10^7 abs. e.m.u./ gm. As the maximum
error admitted in each case was only of order 1 part in 1000, the relatively large
difference between the two values was more than a little disturbing. On the other
hand, Millikan’s value of e (only slightly modified in revisions of the calculations,
made by Birge(1) and by Millikan(2) himself) was generally accepted as accurate
to within 1 part in 1000. In fact it seemed very likely that the electronic charge
lay somewhere between 4.768 and 4.772 x 10 ^ - 10 e.s.u. It is significant of the
authority attaching to Millikan’s work that it was considered necessary to correct
his values for the small difference between absolute and international electrical
units and for a small change in the accepted value of the velocity of light.
The situation has oddly changed since 1929, the spectroscopic and deflection
values of e/m now being in excellent agreement at something very near to 1758 x 10^7.
There are, however, now in the field two values of e which differ by more than
7 parts in 1000; they bear in fact to one another almost exactly the celebrated and
possibly significant ratio 136/137(3). The first of these is Millikan’s, the second is
the value deduced from absolute X-ray wave-lengths by a method which was only
beginning to be fully exploited at the time of Birge’s first paper.
where reference 1 is Probable Values of the General Physical Constants (1929) reference 2 is Millikan's THE MOST PROBABLE 1930 VALUES OF THE
ELECTRON AND RELATED CONSTANTS (1930)
In other words, by 1929 there was data by a different technique that contradicted Millikan's value.
For further information see:
Note on the Value of the Electric Charge (1929)
Absolute Wave-Lengths of the Copper and Chromium K-Series (1931) (finds the e is 4.806 × 10^−10 e.s.u., above the currently accepted/defined value of 4.8032 × 10^−10 e.s.u.)
Viscosity of Air and the Electronic Charge (1935).
The charge of the electron (1937) (a follow up to the 1935 article with the same title)
The Atomic Constants A Revaluation and an Analysis of the Discrepancy (1939)
where the values found by both Bearden and Cork are above the current value.
So in 1931 it was first realized that Millikan's 1930 value disagreed with X-ray technique values due to Millikan's use of an inaccurate viscosity of air value. And this was generally accepted by the 1935-1939 time frame.
In conclusion, there was no hesitation to report correct values in the primary (experimental) literature, at most there was an initial reluctance in the secondary (review/commentary) literature to accept that the values by the new X-ray technique were correct.