The second conjecture, perhaps a more probable one, is that the artifact—presumably a vase—was made by a Chinese artisan inspired by the Koguryo. One look at Wang Hwae Do (circa 627-649 A.D.) gives sound support to this possibility. 
Wang Hwae Do is the Tang Chinese depiction of the envoys from the Three Korean kingdoms of Koguryo, Paekchae, and Silla (Park 376-7). The one on the far left, from Koguryo, wears the perennial bird-feather cholpung that distinguishes him even from the other two Korean envoys. The most markedly Korean attributes, however, end there; we see that the Chinese artist essentially depicted three Chinese men with Korean costumes, with larger, drooping eyes and prominent round nose, and heavier-set body that is distinctly unlike the lithe, slender body type of the Koguryo people. (This is of course the manner in which almost all ancients portrayed foreigners—their own reflected appearance in a foreign garb—and not exclusive to the Chinese). Thus, a Chinese artisan portraying a Koguryo scene will most likely to have painted what we see in the backdrops of the portraits of Baroness Elizabeth and Friederike Maria Beer: figures with Chinese faces wearing indisputably Koguryo attire, especially the headwear. This conjecture is all the more likely if we are to believe Mrs. Beer’s claim that the source of the backdrop was a Korean vase. Ceramics are undoubtedly the most respresentative and widely-admired Korean art form; Western connoiseurs were already very actively collecting them by the turn of the century (Portal 20). Earlier in the 20th century Mr. Honey of the Victoria and Albert Museum wrote of the Korean ceramics:
The best Corean wares were not only original, they are the most gracious and unaffected pottery ever made. They have every virtue that pottery can have…It seems to speak at first of a serenely happy people, and only later in a time of extreme poverty does its graciousness give way to a wild austerity which is admirable in a different way. This Corean pottery, in fact, reaches heights hardly attained even by the Chinese. (qtd McCune 174)
At a time of such growing Western enthusiasm over Korean ceramics, as well as an influx of archaeological finds from ancient tombs, it would not be unlikely that the Chinese artisan with a penchant for peopled scenes painted a Koguryo-inspired one on a vase and passed it off as a highly-prized, antique Korean one.
Of our two possibilities, this is more plausible, perhaps even probable. Even without the testimony of Mrs. Beer on the existence of a vase, we find a possible evidence to the affirmative in the place where we began our search—the paintings themselves. In the Portrait of Rita Munk III (1917-1918) 
there are three detailed motifs to the sitter’s left and right that appear to be a Korean vase on flowers. 
The silhouette is distinctly similar to the one often seen in the most valued of all Korean pottery, the Koryo celadon. As the thirty-three Koryo celadons collected by George Eumorfopoulos (donated to the British Museum in 1911) show, Westerners recognized and appreciated the celadon earlier than perhaps all other Korean artifacts (Portal 17). The Koryo celadon is, of course, something that the Chinese had long known and admired, as a Chinese poem about the “impossibility of imitating the color of the Korean ware” reveals (McCune 176). Could this indicate that the Chinese artisan mimicked the form of the celadon and painted on it a Koguryo scene—anachronistically, no less—to pass it off as the highly prized Korean vase? 
The puzzle pieces to this question are as missing as the painting is unfinished; Klimt died in 1918 quite suddenly before finishing the portrait, leaving no concrete evidence as to his possession of Korean artifacts. We cannot answer our question by studying his diary, letters, or other biographical records. Nor can we search for these artifacts that seemingly vanished upon his decease. As we have seen, however, he did leave some clues along the way: all we need to do is search for them—no where other than in the paintings.