$\renewcommand{\S}{\mathcal{S}}\newcommand{\l}{\langle}\newcommand{\r}{\rangle}\newcommand{\op}{\mathsf{op}}\newcommand{\fin}{\mathrm{fin}}$Recently, I've noticed that the definitions of special $\Gamma$-spaces and spectra are quite close in spirit:

• $\Gamma$-spaces are pointed functors $X\colon(\Gamma^\op,\l0\r)\to(\mathcal{S},*)$ from Segal's category to the category $\mathcal{S}$ of spaces. Moreover, we call $X$
  1. special if $X$ sends coproducts to products;
  2. very special if $\pi_0(X_{\l1\r})$ is a group.
• Spectra are reduced excisive functors $E\colon\mathcal{S}^\fin_*\to\S$ of $\infty$-categories, where
  1. $E$ is excisive if it sends pushouts to pullbacks.
  2. $E$ is reduced if $E(*)\simeq *$;

In particular, very special $\Gamma$-spaces are equivalent to connective spectra. In a separate question, I've asked about whether it's possible to view nonconnectivity as arising from enlarging Segal's category $\Gamma^{\mathsf{op}}\overset{\mathrm{def}}{=}\mathsf{Sk}(\mathsf{FinSets}_*)$ of finite pointed sets into the $\infty$-category $\S^\fin_*$ of finite pointed spaces.

From a different side of this comparison, however, I was also thinking about how we may compare the excision and "special" conditions to each other: indeed, the former implies the latter, and this makes spectra into intrinsically grouplike notions. Because of this and other properties, spectra are regarded as the analogue of $\mathsf{Ab}$ in higher algebra, and the connective ones recover precisely the $\mathbb{E}_\infty$-group objects in spaces.

Question. Is there a known suitable weakening of the excision condition, making it into a kind of "semi-excision condition", in such a way that reduced semi-excisive functors $\S^{\fin}_*\to\S$ ("semispectra") include the $\mathbb{E}_{\infty}$-monoids in spaces as precisely the "connective semispectra"?