I cannot recall that I have ever seen an H with "Only" line engraving. The early ones I have seen, & have, have no engraving on either the plates or top lever. However those with the scroll also have a line border, squiggly, & a top lever which looks just like the one pictured. What I could not decide for sure was if this one has the scroll or if that just some light reflection ahead of the plate screw & below the cocking indicator. If there's scroll there everything else looks proper for a late H & probably right for one in the 57K range.
Note the GE pictured has only straight line engraving on plates with sunburst screw ends & only a line with embellished ends on top lever. My G in the 29K range is identical, except non ejector. The late G's were of course enhanced, but simply do not recall the difference between them & the late H's. I think though the primary difference between the two was always Damascus VS Twist, only slight differences in engraving & possibly a little better grade wood for the G.