Could you send this photo here?
In "Die Habsburger - ein biographisches Lexikon" (I think it was the one by Brigitte Hamann), a quite huge and detalled one, which I had borrowed several times at the German library in Frankfurt there is an article on Archduke Ernst, where the Wallburg children are mentioned. You will certainly know this book, but there the case with the Wallburgs is called "not to be proven". It is said that after the death of Ernst the three "Wallburg descendants" suddenly (the forth had already committed suicide) appeared and wanted to be accpted as legal heirs of Ernst. They tried to prove that Archduke Ernst had been married to Laura von Skublitz (called Baronin Wallburg) and had lived with her in Laibach. Because the (said) marriage was not legitimized by Franz Joseph anyway, the "heirs" did not get any heritage. But the brothers of Ernst paid education money for the (alleged) children of Ernst.
It is also said in the article that there may have been something with false documents, and the whole case does not seem very secure to me, because it is clearly said that "the case could not be clarified". The Wallburgs shall have published two books, one is mentioned in the dictionary:
Ernst Baron von Wallburg, Ein Justizmord...Leipzig,
Other literature, on which the article seems to been based : o.J.; Eugen von Szimics, Die heimliche Ehe des Erzherzogs Ernst von Österreich und seine Kinder, Zürich, 1901.
I think the whole case is strange. It always makes me worried that there are so many cases in history (especially with alleged children, often people, who want money) and cases that cannot been clarified.
It would be interestesting to find more literature, which has an "objective" opinion towards that Walllburg case. Unfortunately until now I haven´t found more, only the article by Hamann.
It´s not proof, but it would be interesting, if you should have any pictures of the children, if they show resemblences to Ernst