An idea occurred to me which I don't think I have had before, though it is so obvious that I will be surprised if I am really the first person to have thought of it.
In Mattingly's Issue 4, as the two dated types show, the mint struck both denarii and antoniniani in 240 (TR P III COS II), but antoniniani only from 241 on, until the end of the issue early in 243 (TR P IIII COS II - TR P VI COS II).
My idea: in 240, the two silver denominations were probably not struck simultaneously, but rather reflect a return to the antoninianus as the sole common denomination in silver, all of the denarii having been struck before that reform and all of the antoniniani thereafter.
The three earliest issuers of antoniniani, namely Caracalla from 215 to early 217, Macrinus briefly at the beginning of his reign, and Elagabalus for a year or so in 218-9, had continued the production of denarii simultaneously with that of the new denomination.
After Elagabalus, however, the two silver denominations seem to have become incompatible: the mint struck one or the other of them as the standard silver denomination, but not both together.
So Balbinus and Pupienus, following the practice of Elagabalus from 219 on, Severus Alexander, and Maximinus Thrax, began their reign striking denarii only, but then discontinued the denarius and struck only antoniniani at the end of their short reign. The reform was signalled not only by the larger module and the reappearance of the radiate crown on the obverse, but by the introduction of new reverse types showing clasped hands with six different legends, none of which of course ever appeared on denarii. About half of Pupienus' antoniniani and some rare bronzes call him MAXIMVS on the obverse, but none of his denarii include that epithet: apparently he decided to add this name to the coins only towards the end of the reign, some time after the antoninianus had supplanted the denarius as the sole regular silver denomination of the two emperors.
Gordian III at first followed the practice of Balbinus and Pupienus: the antoninianus was his only common silver coin in his Issues 1-3 of 238-early 240. At that time, however, Gordian apparently chose to revert to a coinage of denarii only, and struck the issue with TR P III Horseman and the five undated types that Mattingly and Briac erroneously attribute to the following year.
It is unlikely, then, that at the beginning of his next issue (Mattingly's Issue 4) Gordian produced denarii and antoniniani simultaneously: rather he struck the denarii first and then returned to the system based on antoniniani alone.
The Eauze Hoard figures for the successive dated types of 240 give us an approximate idea of how this succession of issues may have been spread across the year:
End of Issue 3, only one dated type (TR P III Emp. sacr.), 15 antoniniani.
TR P III Horseman issue, an average of 27.5 denarii, the equivalent of almost 14 antoniniani, for each of the six types.
Beginning of Issue 4, denarii only, 21 and 12 specimens, the equivalent of about 8 antoniniani, for the two dated types.
Ditto, but antoniniani only, 22 and 16 specimens for the same two dated types.
Assuming a more or less constant rate of silver-coinage production by value rather than by number of specimens, the conclusion of Issue 3, 27% of the production of 240, will have lasted from 10 Dec. 239 until c. middle March 240; the TR P III Horseman issue of denarii (25%), c. mid-March to mid-June; the denarius phase of early Issue 4 (14%), c. mid-June to early August; and finally the antoninianus phase of early Issue 4 (34%), c. early August-10 or 31 December. So the period when Gordian's mint was striking denarii rather than antoniniani as the standard denomination in silver may have lasted from approximately March-August 240.
In the Eauze hoard report, p. 245, the denarii of early Issue 4 are indeed listed before the antoniniani, so the authors might well have thought that they were struck first as I am proposing, though this is not explicitly stated, the treatment of this reign being very brief.